The History of some of Alcuin's Propositiones

Abstract

The Problems to Sharpen the Young [Propositiones ad Acuendos Juvenes] of Alcuin is a major landmark in the history of mathematics in general as well as in the history of recreational mathematics in particular. In this talk I will try to explain and justify this thesis. Some of the reasons for my assertion are general. To the casual observer, the millennium from the fall of Rome to the Renaissance appears as a bleak and indeed Dark Age. But closer observation reveals that major changes begin in Carolingian times - changes which later burst into full development as the Renaissance and the Scientific and Industrial Revolutions. The court of Charlemagne was the centre of the revival of learning in Europe. At the same time, an agricultural and technological revolution made it possible to exploit the rich lands of northern Europe. These events are the direct precursors of the Renaissance and the Scientific Revolution. The Propositiones appears at this time and place, giving us an excellent view of the intellectual life of this period. The Propositiones is in fact the oldest collection of mathematical problems in Latin and really represents the first novel mathematical material to appear in Latin. We shall see that it is a mixture of old and new - there is some appalling ignorance and some ancient problems combined with a truly extraordinary variety of new problems. More specifically, the Propositiones contains 9 to 11 new types of problem, 3 new variants of problems and 2 types which are new to Europe. These comprise 24 to 27 of the 56 problems occurring - a remarkably high percentage of novelty for any collection of this nature and certainly enough to mark it as being of major importance, as shown by the fact that it has been regularly cited in books on recreational mathematics for several hundred years. Consequently, it is very surprising that this MS was not critically edited until the 1970s and that the first translations into modern languages only appeared in the 1990s. (The numbers of novel types and problems vary with who counts and indeed with the time the same individual counts - I gave different numbers three years ago.) Some of the problems have had lengthy developments since the time of Alcuin, occurring in almost every arithmetic/algebra text down to the modern day. I will examine one of his River Crossing Problems - that of the Three Jealous Couples - and describe recent work on it. However, to appreciate Alcuin's role as a focus of development, I will examine several of the problems which were known before Alcuin. In particular, the Hundred Fowls Problem originated in 5C China and spread throughout the literate world in the 8-9C, occurring in India and the Arabic world at about the same time as it appears in Alcuin. Some of the problems are clearly derived from the Roman and Greek traditions, in particular the use of an incorrect Egyptian/Roman formula for the area of a quadrilateral. Some of these appear in Alcuin in novel variants.